zeroless squares

[Henry in Rotherhithe]

For base b, the same argument gives a possible dividing line for k near
log(b)/(log(b)-log(b-1)) which is slightly less than b*loge(b) or
(b-1/2)*loge(b-1/2)+1/2.

Henry Bottomley

> -----Original Message-----
> From: David Wilson
> Sent: 27 February 2005 02:10
> To: seqfan at ext.jussieu.fr
> Subject: Re: zeroless squares
>
> The same naive estimate can easily be generalize to kth powers, giving the
> estimate s(d) = (10^d)^(1/k) - (10^(d-1))^(1/k) for d-digit kth powers.
> p(d) remains the same.  The resulting estimates have ratio
> (9/10)*10^(1/k).
> We should expect an infinite number of zeroless kth powers when this ratio
> is >= 1, which it is for k <= 21.  For k >= 22, the ratio is < 1 and we
> should expect a finite number of zeroless kth powers.
>
> This argument can be extended to other bases, but I don't want to do it.